Invoking [1]} and Ekeland's variational principle [2]} we may assume without loss of generality that \((\mathcal {J}_\infty ^\ell )^{\prime }(\mathbf {u}_k)\rightarrow 0\)  in the dual space \((\mathcal {H}^\ell )^{\prime }\) . From (REF ) and Lions' lemma [2]} we see that there exists \(\theta >0\)  such that
\(\limsup _{k \rightarrow \infty }\sup _{y \in \mathbb {R}^N}\int _{B_1(y)}\mu _i|u_{ik}|^p>2\theta >0\qquad \text{for all \ }i=1,\ldots ,\ell .\) 
